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 ≲ z ≲ 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 ∼ 1100, 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. ∼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. ) allowed significant improvements in the measurements of the intergalactic Hi density. Studying the decrease in flux blueward the Lyman-α (Ly-α) 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(). 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-α absorption lines arising from discrete Hi clouds, was reported for the first time by . Today, the sum of these discrete absorption features is called the Ly-α forest and after the disclosure of its intergalactic origin () 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 [fig:QSOspectrum] shows an example of flux density as a function of wavelength (λ) from the optical spectrum of a QSO at zem = 3.11. The densely distributed, apparently discrete absorption features that constitute the Ly-α forest spread bluewards of the Ly-α emission line (λ= 4996.40Å) down to the Lyman-β (Ly-β) emission line (λ= 4215.71Å) shortwards of which the Ly-α absorption is accompanied by subsequent higher orders of Lyman transitions. Longwards of the Ly-α 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). [image] 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: {c}(,t,})}{\partial t}+}\cdot\nabla I_{\nu}(,t,})=-(,t,})+j_{\nu}(,t,}), where $I_{\nu}(,t,})$ represents the specific intensity that characterizes the source of radiation, c is the speed of light, $(,t,})$ is the attenuation coefficient of the intergalactic medium and $j_{\nu}(,t,})$ 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 R represents the rate at which the energy, carried by photons of frequency ν in the direction $}$, crosses a unit area per unit solid angle per unit time (). In the case of a single background source, such as a QSO, jν = 0 and the solution of Equation [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: (,t,})=\rho(,t)(,t,})+n(,t)(,t,}), where ρ(R, t) is the mass density of the gas, $(,t,})$ is its opacity, n(R, t) is the number density of scattering particles of a specific mean mass $$ and $(,t,})$ is the scattering cross section (). The absorption features arise from the scattering of photons traveling from the background quasar through an intergalactic medium with number density n(R, t). The resonance line scattering cross section will depend on the specific characteristics of the transition considered (rest-wavelength λ₀ or frequency ν₀ 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: =\left(}{m_{e } c}\right){4\pi} f(p,\nu), where $(p,\nu)={\pi^{1/2}\Delta}H(p,x)$ is the normalized Voigt profile defined by the Voigt function H(p, x)[1] and the Doppler width ΔνD = ν₀b/c. The Doppler parameter b is generally defined with its thermal component $b_{th}=\left(T}{m}\right)^{1/2}$ where kB is Boltzmann’s constant and m is the mass of the atoms, but a kinematic component (bkin) can be added to take into account possible turbulence effects (e.g. ). The optical depth to absorption at the frequency ν, due to the presence of intervening gas along a line of sight (at the position s′ and time t′) from the source position (s₀) to a position s, is then given by the solution of Equation [eq:radTrans] : =}^{s}ds'n(s',t') \end {equation} 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^{-}$. 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 H{\sc \,i} (Chapters 2-3-4). Nevertheless, the H{\sc \,i} Lyman-$\beta$ transition will also be considered in Chapter 4 and the He{\sc \,ii} Lyman-$\alpha$ transition will be used in Chapter 5. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Since the first discovery, the sum of the absorption lines arising from Ly-$\alpha$ absorption in diffuse H{\sc \,i} 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 (). Such a distribution, called Ly-$\alpha$ forest (), suggested that these features arise in distinct localized regions of the IGM, implying the presence of an inhomogeneous density field (). 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 H{\sc \,i} content (e.g.~; ; ; ; ). The properties of the Ly-$\alpha$ absorption systems and their classification are discussed below. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 ($$) given by the sum of the individual optical depths, $(i)$, corresponding to locally overdense regions (): \begin {equation} =(i)=\pi^{1/2}\langle(p,x)\rangle where $={\pi^{1/2}}$ is averaged over the line of sight, weighted by the density and τ₀ is the optical depth at line centre defined as: ={\pi^{1/2}b}=\pi^{1/2}}{m_{e}c}\left[{4\pi}\right]{b}f as a function of the column density N. For the Hi Lyα (λ0α = 1215.67Å, fα = 0.4164) Equation [eq:linecenter] becomes: \sim0.38\left(})}{10^{13}^{-2}}\right)\left({20 ^{-1}}\right)^{-1} The Lyα 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 ≳ 5.5) where the individual lines merge together forming the effect of a trough in the spectrum (). 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, Wλ is usually defined in wavelength notation as: W_{\lambda}=}{c}\int(1-e^{-}) d\nu . For a Voigt profile, Wλ 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 [fig:EqW] is presented a reproduction of the curve of growth for hydrogen Ly-α; it is possible to distinguish three different segments, related to different line profiles (,): - The linear part: in optically thin regions (τ ≪ 1) 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. Wλ ∼ 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 $W_{\lambda}\sim N)}}$. 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 $W_{\lambda}\sim )}}$. [Reproduction of Figure 3 from . The curve of growth for the Ly-α 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: Lyα forest (blue), LLS (yellow), DLA(red). ] The classification of Ly-α 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 [fig:EqW]. Historically, absorption systems with logN(Hi)≲17.2 are called Ly-α forest, those with 17.2≲ logN(Hi)<20.3 Lyman limit systems (LLSs) and those with logN(Hi)≥20.3 damped Lyα absorbers (DLAs) (e.g. ; ; ,; ; ). The number of systems per unit of redshift has been found to decrease as their column density increases. That is, the Ly-α forest absorbers are the most common, they mainly contain ionised hydrogen and may be associated with metals absorbers (e.g. ; ). 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. ). 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.; ; ). 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. ; ). On the theoretical side, models to reproduce synthetic Ly-α 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-α forest arise naturally from the fluctuations of the continuous medium formed by the gravitational collapse of the initial density perturbations (e.g. ; ). 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-α forest absorption systems (e.g. ). According to these models, most of the volume of the Universe is occupied by low density gas with a $\rho / \lesssim 10$, where $$ 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-α 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-α 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 [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. [A summary of the main phases of evolution of the IGM, from the neutral gas that filled the Universe after the recombination to the reionized gas that fills the space between galaxies after the three reionization events driven by stars and quasars. The ionization potentials for the different transitions are also reported.] Hydrogen reionization From the study of the cosmic microwave background anisotropy it seems that not later than z ≃ 11 () the UV radiation emitted by the first objects, mostly stars (), 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-α absorption features of several quasars have shown that the intergalactic hydrogen was completely reionized by z ≃ 6 (e.g. ; ; ). This redshift corresponds to a rapid rise of the Hi Ly-α optical depth ($}}$), 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 $}}$, this interpretation is not unique (). Moreover, some evidence for the final stages of a patchy hydrogen reionization has been found recently in at z ∼ 6, suggesting a later end of this process (at z ∼ 5). If for z ≳ 6 the Ly-α 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 ∼ 3, 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 (). Theoretically, their much harder photons would have been able to fully ionize Heii at a redshift 3 ≲ z ≲ 4.5, but these estimates change depending on assumptions about the abundance of QSOs and the hardness of their spectra (). Direct observation The most direct evidence for Heii reionization derives from measurements of the Heii Lyman-α optical depth ($}}$). The evolution of $}}$ 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. ; ). The latter is known as the “Gunn–Peterson absorption" (). 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-α line (304 Å) and the density fluctuations in the IGM, the Heii Gunn–Peterson trough is sensitive to ion fractions $x_{ }}\gtrsim 0.01$, while the hydrogen one saturates at $x_{}}\simeq 10^{-5}$ (; ). In addition, luminous quasars create large fluctuations in the ionizing background, making the flux transmission possible even during the early stages of Heii reionization (). The intergalactic Heii Lyman-α 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-α 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 ≃ 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 ≳ 3 (; ), whereas the absorption becomes patchy at z ≲ 3, reflecting an intermediate phase of reionization, and evolves into Heii Ly-α forest (discrete absorption lines) at z ≲ 2.7 (e.g. ; ). 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” () and the discovery of 2 new ones at z < 3 (). But even if in these recent works the end of the Heii reionization seems to be observed at z ≃ 2.7 (e.g. ; ), 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 ), 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-α forest. Variation of metal-line ratios (like Civ/Siiv; ) 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 (). These measurements generally agree with a z ≃ 3 reionization () 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-α forest. In particular, one of the effects of an injection of a substantial amount of energy could have been a lower average Hi Ly-α opacity (; ). However, a small “dip” in the Hi Ly-α opacity observed at z ≃ 3.2 () does not have a straightforward interpretation (; ) and has not been confirmed in recent and refined measurements (). 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 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 (). At different redshifts, the thermal state of the intergalactic medium can be described through its temperature–density (T–ρ) relation. In the simplest scenario, for gas at overdensities Δ ≲ 10 ($\Delta=\rho / $, where $$ is the mean density of the IGM), cosmological simulations show a tight power-law relationship between temperature and density: T(\Delta)=T_{0}\Delta^{\gamma -1} \ , where T₀ is the temperature at the mean gas density (; )). The reason why the same power-law holds for a vast range of overdensities (Δ ∼ 0.1 − 10) has been explored in different works (e.g. ; ) and can be intuitively attributed to the balance between photoheating by the UV background and the adiabatic cooling due to Hubble expansion (but see also ). It has been shown that this relation is likely to be established in Δz ∼ 1 − 2 after the end of each of the reionization events (e.g. ; ; ; ). While the dynamics of the reionization processes are still partially unclear, constraining the evolution of the parameters T₀ and γ as a function of redshift will describe, with high precision, the evolution of the IGM thermal state, and bring important insights about these events. Different cosmological simulations, with different prescriptions for heating and radiative mechanisms, predict that, during the reionization, the parameters T₀ and γ can undergo variations that, if identified, can robustly characterize the timing and physical mechanisms that drove the evolution of the comic gas (e.g. ; ; ). While in principle the evolution of the IGM T–ρ relation can be studied for both the reionization epochs, in practice the Heii reionization at z ∼ 3 is the only one really accessible to the constraint using quasar spectra: the strong blending of the absorption features and the lower quality of the spectra do not allow yet a reliable analysis at z ≳ 5 (but see ). For this reason the work presented here is focused on the redshift range spanning the second reionization event (1.5 ≲ z ≲ 3.8). The IGM thermal state during the Heii reionization It is predicted that the IGM is reheated to several 10⁴ K by photo-heating during the completion of the Heii reionization at z ≃ 3, leaving an ‘imprint’ on its temperature evolution. This boost in temperature is predicted in the case in which the UV photons that ionize the neutral gas are absorbed with higher energy than the ionization potential of the atoms, and so the free electrons will be released with some energy to share with the surrounding baryons through Coulomb scattering (e.g. ). The evolution of the parameter T₀ In Figure [fig:T0evolution] is presented a qualitative prediction for the evolution of T₀ during the Heii reionization driven by the UV radiation emitted by quasars, motivated by several models of thermal thermal state evolution (e.g. ; ; ). Before the reionization, the condition of balance between photoheating and cooling, mainly due to the adiabatic expansion of the Universe, results in a thermal asymptote, completely determined by the shape of the ionizing spectrum (). Generally, quasar ionizing spectra are harder than stellar spectra and this is reflected in a higher temperature asymptote. At the mean density the characteristic signature of reionization is expected to be a peak: a marked heating followed by a subsequent cooling due to adiabatic expansion at the end of which the temperature of the gas will return to follow a new asymptotic decrease. If quasars dominate the UV background (UVB) after the Heii reionization, the expectation is to be able to distinguish a shift toward higher temperatures in the new thermal asymptote at z ≲ 2. A precise constraint on the evolution of T₀ is therefore of fundamental importance because it provides timing of the reionization event and also possible information about its sources. [Qualitative predictions for the possible evolution of the temperature at the mean gas density during the Heii reionization. Top panel: schematic representation of the IGM reionization process, from the overdense regions, where we expect quasars (black symbols) to switch on, the ionizing bubbles (yellow patches) expand and overlap, completing the reionization of the entire medium. Bottom panel: expected evolution of the temperature at the mean gas density during the reionization process; after a temperature peak, the cosmic gas is expected to return to follow a thermal asymptote determined by the hardness of the ionizing spectrum of the sources dominating the UVB.] The evolution of the parameter γ The evolution of the parameter γ that defines the slope of the IGM temperature–density relation (Equation [eq:TDrelation]) can be related with the manner in which the photoionization fronts expand and so with the topology of the reionization process (e.g. ; ; ). In the condition of balance between photo-heating and cooling due to adiabatic expansion, assumed to apply both before and after the reionization event, the value of γ is predicted to assume the asymptotic value of γ ∼ 1.6 (). A T–ρ relation with a positive slope (i.e. γ > 1) corresponds to a situation in which higher overdensities present higher temperatures because they are more bounded against the expansion and experience higher recombination rates (that will result in more atoms for the photoheating). Depending on the topology of the Heii reionization γ could assume different values. So far predictions from cosmological simulations about how and if this variation would happen rely on specific prescriptions of photoheating and radiative transfer mechanisms (e.g. ; ; ). One often-proposed possibility is presented in Figure [fig:gammaevolution] (e.g. ; ). According to this picture, if the reionization proceeds from high density regions (where quasars switched on) to low density ones, then the voids will be the last to reionize and reheat and, immediately after the end of the reionization, they will suffer the least amount of cooling and hence they will contain hot gas. It is then possible that, for a small redshift range, before the underdense regions cool down again, the IGM T–ρ relation approaches an isothermal value (γ ∼ 1). The study of the evolution of γ in the redshift range spanning the reionization is therefore particularly interesting for understanding the physical process governing this event. [Qualitative predictions for a possible evolution of the parameter γ during the Heii reionization. Top panel: schematic representation of the IGM reionization process, from the overdense regions, where we expect quasars (black symbols) to switch on, the ionizing bubbles (yellow patches) expand and overlap, completing the reionization of the entire medium. Bottom panel: expected evolution of the parameter γ during the reionization; changes from the asymptotic values of γ ∼ 1.6 can be expected during the reionization depending on the topology of the process.] Previous methods to measure the equation of state Temperature variations affect the structures of the Hi Ly-α forest and observations of this region of quasar spectra have been considered the best method to obtain information on the IGM evolution. The widths and depths of the Ly-α lines are mainly set by the column densities of the absorbers, Hubble broadening as light travels across the absorbing gas, peculiar velocities and thermal broadening (e.g. ). Furthermore, recently it was shown that the broadening of these features is not only affected by the instantaneous temperature of the gas at the time of absorption but also indirectly by its thermal history at earlier times. This effect is referred as _Jeans smoothing_ and reflects changes in the density distribution of the IGM on small scales (; ; ; ; ). Temperature information can be extracted from the analysis of the absorption line shapes but this process is not straightforward and complex cosmological simulations are required to characterize the large-scale structure and bulk motion of the IGM. For this reason any measurement of the IGM thermal state from the Ly-α forest, has been always strictly entangled in state of the art cosmological simulations that, while allowing a better understanding of the physical processes involved, introduce possibly relevant systematic uncertainties that have to be taken into account. Previous efforts can be divided into two main approaches: the study of individual absorption features and the quantification of the absorption structures with a global statistical analysis of the entire forest. Line-profile analysis The first method consists of decomposing the Ly-α spectra into a set of Voigt profiles. Since the minimum line width (b-parameter) can be considered dominated by thermal broadening, and since the column density (N) correlates strongly with the physical density, it is possible to trace the T–ρ relation (Equation [eq:TDrelation]) using the low-b edge of the b(N) distribution (; ; ; ; ; ). Even if simulations are needed to properly calibrate the relationship between the temperature–density relation and the b(N) distribution, this technique represents the most direct test to simultaneously constrain the value of both the parameters T₀ and γ. Nevertheless, the results obtained so far are difficult to interpret because of the large statistical uncertainties: and found some evidence for an increase in the IGM temperature and a decrease in the value of γ at z ≃ 3, consistent with some expectations of Heii reionization while, in contrast, using similar data, found a constant temperature over z ∼ 2 − 4. Recently, the most precise measurement with this method has been presented in at z ≃ 2.4. In their work they recalibrated with hydrodynamical simulations the line-profile analysis of , computed using a large sample of Hi absorbers (∼6000), and found a value of γ = 1.54 ± 0.11. Statistical analysis of the transmitted flux While line fitting can be a time consuming and somewhat subjective process, a characterization of the transmitted flux using global statistical approaches, without decomposing it into individual absorption features, represents a much easier way to analyze the forest and extract information from the comparison with theoretical models (e.g. ). Examples of this general approach are the study of the flux probability distribution (PDF) based on pixel statistics (e.g. ; ) and the wavelet analysis method (e.g. ; ). The Ly-α flux probability distribution function represents the simplest pixel statistic sensitive to the density distribution and the thermal state of the IGM. However, previous measurements of the parameter γ using this method have provided contradictory results: while in the recent work of using a large sample of 3393 spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) they found that a γ = 1.6 best describes the data over a redshift range z = 2.3 − 3.0, previous measurements seemed to favor an ‘inverted’ temperature–density relation (γ < 1). The work of first suggested that a γ < 1 may provide a better fit to the data at low redshifts (z < 3). In this scenario, lower density regions are hotter, an interpretation that seems at first to be counterintuitive because denser gas is actually expected to trace regions more bounded against the expansion of the Universe, in which the adiabatic cooling is suppressed and the recombination is faster, yielding more neutral atoms per unit time for photo-heating. Evidence of a possible inverted T–ρ relation was also reported in other analyses (; ; ; ) and a possible explanation was suggested by considering radiative transfer effects (). Although it appears difficult to produce this result considering only Heii photo-heating by quasars (), a new idea of volumetric heating from blazar TeV emission seems to naturally explain an inverted temperature–density relation at low redshifts (; ; ). The current uncertainties in the constraints on the T–ρ relation through the PDF can be related to the sensitivity of this technique to a range of systematic uncertainties, among which the contamination by metal absorption lines have the major impact in high resolution and high signal-to-noise (S/N) spectra (). Indeed, the uncertainties remain substantial using the wavelet decomposition, that involves the filtering of spectra to obtain the amplitude of the signal, which is in-turn sensitive to variations in the temperature: and found evidence for Heii reionization completing near z ∼ 3.4, but with large statistical uncertainties (1σ > 20per cent). In the recent work of , the PDF and the wavelet decomposition methods have been compared and tested on observed Ly-α spectra at low redshift: even if the results are in formal agreement with previous measurements, the uncertainties are still large and there is a mild tension between the two analysis, with the flux PDF measurements generally preferring a lower value for γ. A summary of the existing γ and T₀ measurements is presented and discussed in Chapter 6 (see Figure [fig:finals]) Much more robust measurements can in principle be obtained with the curvature statistical approach, used for the first time by . This statistic represents the fulcrum of the analysis provided in this work, and its strengths, weaknesses and developments will be extensively discussed in the following Chapters. Thesis aim In the scenario described in this Introduction, a general lack of knowledge about the behavior of the T–ρ relation in the redshift range spanning the Heii reionization emerges. A further investigation of the evolution of the IGM thermal state during this phase assumes extreme importance in order to find constraints on the physics of the reionization process and on its effects on the intergalactic gas. Indeed, a better understanding in this field will bring important information about the radiation sources, the heating mechanisms and the phenomena that characterize the structure of the intergalactic medium in the latest chapter of its evolution. Therefore, this thesis intends to obtain a better constraints on the thermal state of the IGM in the redshift range z ≃ 1.5 − 3.8 using a combination of high quality observational quasar spectra and state of the art hydrodynamical simulations. The general approach adopted here aims to study, statistically, the shapes of the features arising from the absorption by the intergalactic gas and, using the comparison with synthetic spectra, extract information about the evolution of the IGM temperature–density relation. The statistic extensively used in this work is the _curvature_, introduced recently in the Ly-α forest analysis by . The curvature statistic is able to deliver the most precise measurements of the IGM temperature at the gas overdensities traced by the Ly-α forest. However, probing only a narrow density range at each redshift, the curvature so far has not been used to constrain the parameter γ of the T–ρ relation. For this reason, the investigation presented in this work will not be limited in the use of the curvature to find temperature constraints but it will also explore a “companion" statistic that should in figure allow precise measurements of γ in order to obtain a complete characterization of the IGM temperature–density relation. Thesis Structure A detailed description of the curvature method, and of the procedures necessary to apply this statistic to the available observational spectra and suite of hydrodynamical simulations is presented in Chapter 2. After the selection and preparation of a uniform sample of high quality Ly-α forest sections, synthetic spectra have been created and calibrated to match as closely as possible the characteristics of the real ones. A correct match of the noise properties and optical depth values between real and simulated spectra is shown to be particularly important in order to obtain reliable values for the characteristic overdensities ($$) probed by the Ly-α transition, at each redshift (see also Appendix A). Chapter 3 describes and discusses the new measurements of the IGM temperature at the characteristic overdensities ($T()$) probed by the forest. The temperature values at the mean gas density, T₀, are also obtained from the $T()$ under the assumption of different γ values. Because the curvature method, applied only to the Ly-α forest region, does not allow an independent γ constraint, new statistics have to be used to obtain a complete characterization of the T–ρ relation. The results of Chapter 2 and Chapter 3 have been published in . Chapter 4 presents a new development of the curvature statistic that allows to obtain constraints on the parameter γ. This new statistic, _the curvature ratio_, represents, at each redshift, the ratio between the curvature computed from the Hi Ly-β forest (contaminated with lower redshift Ly-α absorption) and the corresponding Ly-α forest along the same line of sight to a quasar. Because the overdensities contributing to the Ly-β absorption are, on average, higher than that for the Ly-α, the curvature ratio incorporates, at each redshift, information about the temperatures of two different gas densities and so will be sensitive to γ. The curvature ratio method is applied to our observational and simulated samples and the strengths and weakness of this method are explored and discussed. The preliminary results of Chapter 4 have been submitted for publication to Monthly Notice Letters of the Royal Astronomical Society but a more substantial description of our analysis is presented in this thesis and will be characterized by the text preceded by the symbol “*". Chapter 5 describes the preliminary results of an investigation aiming to understand the feasibility of future applications of the curvature statistic to the Heii Ly-α forest. Being aware of the current observational limitations in the number and quality of the available spectra, it is also shown that any Heii Ly-α forest curvature analysis is currently prevented by the requirements of hydrodynamical simulations necessary to accurately resolve the helium absorption. The results of Chapter 3, 4, and 5 are summarized and combined in Chapter 6 for a final discussion about the thermal state of the IGM that is the main focus of this thesis. The results of this investigation will be the starting point for new projects that will be the natural extension of the research shown here. [1] $H(p,x)={\pi}^{+\infty}dy}}{(x-y)^{2}+p^{2}}$ with p = Γ₀/(4πΔνD) and x = (ν − ν₀)/ΔνD. Γ₀ is the radiation damping width of the transition (e.g. Γ₀ = 6.262 × 10⁸s−1 for the Hi Ly-α) .