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The curvature ratio

\label{chp:The curvature ratio}

The post-reionization thermal state of the intergalactic medium is characterized by a power-law relationship between temperature and density, with a slope determined by the parameter \(\gamma\). We describe a new method to measure \(\gamma\) using the ratio of flux curvature in the Lyman-\(\alpha\) and \(\beta\) forests. At a given redshift, this curvature ratio incorporates information from the different gas densities traced by Lyman-\(\alpha\) and \(\beta\) absorption, thereby breaking the degeneracy between \(\gamma\) and the temperature inferred at the gas mean density. It is relatively simple and fast to compute and appears robust against relevant sources of observational uncertainty. We apply this technique to a sample of 27 high-resolution quasar spectra from the Very Large Telescope, finding preliminary results broadly consistent with \(\gamma\sim 1.5\) over the redshift range \(z\sim 2.0\)\(3.5\). However, while promising statistical errors appear to be achievable in these measurements, uncertainties in the assumptions about the thermal state of the gas and its evolution may complicate this picture.

Boera, E., et al. 2015, MNRAS Letters, submitted

The results presented in Chapter 3 highlight the necessity of a precise constraint of the parameter \(\gamma\) of Equation \ref{eq:TDrelation} to completely characterize the IGM thermal evolution shown in Figure \ref{fig:slopes} in a model-independent way. Therefore, the second part of this thesis will be dedicated to this aim.

During and immediately after the H i and He ii reionisations (citation not found: McGreer15) (citation not found: Syphers14) (citation not found: Worseck14) cosmological simulations predict that \(\gamma\) may vary, becoming multi-valued and spatially-dependent according to the dynamics, heating and radiative transfer mechanisms involved (citation not found: Bolton04) (citation not found: McQuinn09) (citation not found: Meiksin12) (citation not found: Compostella13) (citation not found: Puchwein15). Despite considerable recent improvements, accurately simulating the effect of reionisation events on the IGM remains an open challenge. Furthermore, current measurements offer a somewhat confusing observational picture.

The main laboratory to detect variations in the \(T\)\(\rho\) relation has been the H i Lyman-\(\alpha\) forest in quasar spectra. Efforts to infer the thermal state of the IGM and search for signals of reionisation have used either line-profile decomposition to measure gas temperature as a function of column density (citation not found: Schaye00) (citation not found: Ricotti00) (citation not found: McDonald01) (citation not found: Rudie13) (citation not found: Bolton13) and a variety of statistical approaches which are valuable at higher redshifts, \(z>3\), where line fitting is problematic (citation not found: Theuns02) (citation not found: Becker07) (citation not found: Bolton08) (citation not found: Lidz10) (citation not found: Becker11) (citation not found: Boera14). While these methods probe wide redshift and density ranges (\(z\approx 1.6\)–5, \(\Delta\approx 0.3\)–8), large uncertainties in the measurements of \(T_{0}\) and \(\gamma\) may be caused by strong degeneracies between the effects of temperature and density on Ly-\(\alpha\) forest absorption.

One way to reduce these degeneracies is to constrain the \(T\)\(\rho\) relation by comparing Ly-\(\alpha\) and higher-order Lyman-series transitions, such as Ly-\(\beta\). Ly-\(\beta\) lines of moderate optical depth (\(\tau\sim 0.1\)–1.0) arise from higher overdensities at which Ly-\(\alpha\) lines are saturated. That is, the Ly-\(\alpha\)-to-\(\beta\) optical depth ratio is \(f_{\alpha}\lambda_{\alpha}/f_{\beta}\lambda_{\beta}=6.24\) (proportional to the ratios of oscillator strengths and rest wavelengths). Statistically comparing Ly-\(\alpha\) and \(\beta\) absorption is therefore a promising approach for measuring \(\gamma\). Indeed, using the Ly-\(\beta\) forest in IGM temperature measurements has been suggested in several theoretical works (citation not found: Dijkstra04) (citation not found: Furlanetto09) (citation not found: Irsic14). However, so far no practical attempt has been made to directly measure \(\gamma\) from a joint Ly-\(\alpha\) and \(\beta\) forest analysis. One challenge is that the Ly-\(\beta\) forest (the region between the Ly-\(\beta\) and Ly-\(\gamma\) emission lines) is entangled with absorption from Ly-\(\alpha\) at lower redshifts and it is difficult to separate the two. Hereafter we will refer to the total Ly-\(\beta\) plus the foreground lower redshift Ly-\(\alpha\) absorption as the Ly-\(\beta\)\(+\)\(\alpha\) region. However, assuming that these Ly-\(\beta\) and \(\alpha\) lines arise from physically uncorrelated IGM structures, a possible strategy to overcome this problem is to statistically compare the properties of the Ly-\(\beta\)\(+\)\(\alpha\) and corresponding Ly-\(\alpha\) regions.

In this Chapter we present a new method to constrain the slope of the \(T\)\(\rho\) relation using the two forest regions (Ly-\(\alpha\) and Ly-\(\beta\)\(+\)\(\alpha\)) in 27 high resolution quasar spectra. We use a statistic based on the flux curvature analysis of (citation not found: Becker11) and Chapters 2 & 3. These previous works demonstrated that the curvature method can measure the temperatures at the (redshift dependent) characteristic densities probed by the Ly-\(\alpha\) forest. However, as only a narrow density range is constrained, it has not yet been used to measure the slope of the \(T\)\(\rho\) relation (but see (citation not found: Padmanabhan15) for a recent theoretical analysis using the Ly-\(\alpha\) forest). Using hydrodynamical simulations, we show that, at each redshift, the ratio between the curvatures of corresponding Ly-\(\alpha\) and Ly-\(\beta\)\(+\)\(\alpha\) forest regions (where in the Ly-\(\beta\)\(+\)\(\alpha\) region the redshift always refers to the Ly-\(\beta\) absorption) is sensitive to differences in the IGM thermal state between the two density regimes. Averaged over many lines of sight, this curvature ratio appears to allow \(\gamma\) to be measured with little sensitivity to \(T_{0}\). We demonstrate the potential for this technique using 27 quasar spectra spanning the He ii reionisation redshift range. That potential is currently limited by assumptions underpinning the available simulation suite; we propose how these limitations can be overcome with future simulations and a refined data analysis approach.

The Chapter is organized as follows. In Section LABEL:sec:data is presented the selection of the subsample of quasar spectra used for this analysis from the larger sample of Table \ref{table:datatable}. The production of synthetic forest spectra from hydrodynamical simulations is explained in Section \ref{sec:sim}. The curvature ratio statistic and the main steps of the analysis are introduced in Section \ref{sec:method}. Possible systematic uncertainties in the evolution of the IGM thermal state are extensively discussed in Section \ref{sec:SystErr}. Our preliminary measurements of \(\gamma\) are presented in Section \ref{sec:gamma}. Finally, the results are summarized and interpreted in Section \ref{sec:conclusions2}.

The observational data


The 27 quasar spectra were originally retrieved from the archive of the Ultraviolet and Visual Echelle Spectrograph (UVES) on the Very Large Telescope (VLT). They were selected on the basis of quasar redshift, wavelength coverage and signal-to-noise ratio from the sample of 60 spectra used in Chapters 2 and 3 for measuring the \(z=1.5\)–3 Ly-\(\alpha\) forest curvature. They have resolving power \(R\sim 50000\) and continuum-to-noise ratio \(\geq\)24 pix\({}^{-1}\) in the Ly-\(\alpha\) forest region (see Section \ref{sec:obs} for details). This level of spectral quality is necessary so that the curvature measurement is not dominated by noise and unidentified metal lines. Because our new method compares the curvature of the Ly-\(\alpha\) and \(\beta\)\(+\)\(\alpha\) forest regions, we extended these same criteria to the Ly-\(\beta\)\(+\)\(\alpha\) region at \(z=2.0\)–3.5, reducing the sample from 60 to 27 spectra. The quasar sample details are provided in Table \ref{table:subsample}.

\label{table:QSO}List of the QSOs subsample used for this analysis. Column 1 shows the quasar names and column 2 shows their emission redshifts.
QSO \(z_{em}\)
J110325\(-\)264515 2.14
J012417\(-\)374423 2.20
J145102\(-\)232930 2.21
J024008\(-\)230915 2.22
J212329\(-\)005052 2.26
J000344\(-\)232355 2.28
J045313\(-\)130555 2.30
J112442\(-\)170517 2.40
J222006\(-\)280323 2.40
J011143\(-\)350300 2.41
J033106\(-\)382404 2.42
J120044\(-\)185944 2.44
J234628\(+\)124858 2.51
J015327\(-\)431137 2.74
J235034\(-\)432559 2.88
J040718\(-\)441013 3.00
J094253\(-\)110426 3.05
J042214\(-\)384452 3.11
J103909\(-\)231326 3.13
J114436\(+\)095904 3.15
J212912\(-\)153841 3.26
J233446\(-\)090812 3.31
J010604\(-\)254651 3.36
J014214\(+\)002324 3.37
J115538\(+\)053050 3.47
J005758\(-\)264314 3.65
J014049\(-\)083942 3.71

To establish an initial continuum in the Ly-\(\beta\)\(+\)\(\alpha\) region, we applied the same automatic, piece-wise polynomial fitting approach used in Section \ref{sec:obs} for the Ly-\(\alpha\) region, leaving the latter unchanged. While manual refitting was necessary in some spectra for particular parts of the Ly-\(\beta\)\(+\)\(\alpha\) region, this initial continuum has little effect on the final curvature measurements because the spectra are subsequently “re-normalized” using a b-spline fit, as explained in Section \ref{sec:simA} below, and as in Section \ref{sec:obs}. Similar to Figure \ref{fig:histo}, the redshift and the C/N distributions of the Ly-\(\beta\)\(+\)\(\alpha\) region of the quasars in our sample is shown in Figure \ref{fig:histo2}.

\label{fig:histo2}Histograms showing our sample of quasar spectra with C/N\(>24\) in the Ly-\(\alpha\) and Ly-\(\beta\)+\(\alpha\) forest region. Top panel: redshift distribution of Ly-\(\beta\) in the Ly-\(\beta\)+\(\alpha\) region. Bottom panel: distribution of the continuum to noise ratio (C/N) in the same region (see also Figure \ref{fig:histo} for the equivalent information for the Ly-\(\alpha\) forest). The redshift bins of the histograms have been chosen for convenience of \(\delta z=0.025\). The vertical lines divide the histograms in the redshift bins of width \(\Delta z\sim 0.6\) in which the measurements (at \(2\lesssim z\lesssim 3.8\)) will be collected in the following analysis.

The simulations


We used the same hydrodynamical simulations used in Chapters 2 and 3 to produce synthetic spectra for Ly-\(\beta\)\(+\)\(\alpha\) and Ly-\(\alpha\) in the redshift range \(z=2.0\)–3.5. The gadget-3 smoothed-particle hydrodynamics simulations include dark matter and gas, with \(2\times 512^{3}\) particles and a gas particle mass of \(9.2\times 10^{4}h^{-1}M_{\odot}\) in a periodic box of 10 comoving \(h^{-1}\) Mpc (see Section \ref{sec:sims} for details). The gas, assumed to be optically thin, is in equilibrium with a spatially uniform UVB (citation not found: Haardt01), but the photoheating rates have been rescaled so that the corresponding values of \(T_{0}\) and \(\gamma\) vary between different simulations. The parameters characterizing the different models are summarized in Table \ref{table:simulations}. From each model, synthetic Ly-\(\alpha\) forest spectra were generated for 1024 random lines of sight through each of 6 redshift snapshots over the range \(z=2.0\)–3.5. The Ly-\(\beta\)\(+\)\(\alpha\) (optical depth) spectra were produced by scaling the Ly-\(\alpha\) optical depths by a factor \(f_{\alpha}\lambda_{\alpha}/f_{\beta}\lambda_{\beta}=6.24\) and contaminating them with Ly-\(\alpha\) absorption from lower-redshift snapshots (\(z=1.6\)–2.7) with the same thermal history.

Finally, as in Section \ref{sec:calibration}, we calibrated the synthetic spectra to match the properties of the real spectra (i.e. resolving power, pixel size and signal-to-noise ratio). The only difference is that, in this work, we scaled the effective optical depth of the synthetic spectra to match the recent results from (citation not found: Becker12) rather than to a direct measurement from our spectra. As shown in Section \ref{sec:simA}, our measurements are relatively insensitive to this optical depth calibration.

The curvature ratio method


The curvature statistic is defined as in Equation \ref{c}, with the first and second derivatives of the flux (\(F^{\prime}\), \(F^{\prime\prime}\)) taken with respect to wavelength or relative velocity. As demonstrated in previous works, the Ly-\(\alpha\) forest curvature is directly related to the IGM temperature at the characteristic overdensities probed by this absorption, regardless of \(\gamma\). Because the median overdensity contributing to Ly-\(\beta\) forest absorption is higher than that for Ly-\(\alpha\) (citation not found: Furlanetto09), the mean absolute curvature computed from sections of Ly-\(\beta\)\(+\)\(\alpha\) forest will be, on average, a tracer of the IGM temperature in a higher density regime. Therefore, at each redshift \(z\), the curvature ratio,

\begin{equation} \label{R} \label{R}R_{\kappa}(z)\equiv\frac{\langle|\kappa_{\beta+\alpha}(z)|\rangle}{\langle|\kappa_{\alpha}(z)|\rangle}\,,\\ \end{equation}

will be sensitive to temperature differences between two different gas density regimes and, consequently, to \(\gamma\). Here the redshift \(z\) in the Ly-\(\beta\)\(+\)\(\alpha\) forest region always refers to the Ly-\(\beta\) absorption. In Equation \ref{R} the mean absolute curvatures for Ly-\(\beta\)\(+\)\(\alpha\) and Ly-\(\alpha\) are averaged over corresponding spectral sections of 10 comoving \(h^{-1}\) Mpc (corresponding to the simulation box size). We demonstrate in Appendix B the numerical convergence of our simulations for the curvature ratio in terms of box size and mass resolution.

Analysis of simulated spectra: the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation


We find the connection between the curvature ratio and \(\gamma\) at the redshifts of the simulation snapshots (\(z\)=[2.173, 2.355, 2.553, 2.760, 3.211, 3.457]) by computing the mean \(R_{\kappa}\) over the 1024 artificial lines of sight, for each thermal history, and fitting a simple function between \(\log\langle R_{\kappa}\rangle\) and \(\gamma\).

Single \(R_{\kappa}\) values for each line of sight and redshift were obtained as defined in Equation \ref{R}. To compute the mean absolute curvature of synthetic Ly-\(\alpha\) and Ly-\(\beta\)\(+\)\(\alpha\) sections, we adopt the same procedure described in (citation not found: Becker11) and in Section \ref{sec:curvature}, an example of which is shown in Fig. \ref{fig:Ksim}: in each section of artificial spectrum, the mean absolute curvature is computed from a b-spline fit which has been ‘re-normalized’ by its maximum value in that interval. This approach is required to avoid systematic errors when determining the curvature from real spectra, so it must be applied to the synthetic spectra for consistency. The b-spline fit reduces the sensitivity of \(\kappa\) to noise and the re-normalization minimizes potential uncertainties arising from inconsistent continuum placement. Finally, only the pixels where the re-normalized b-spline fit falls in the range 0.1–0.9 are used to measure the mean absolute curvature of each section. In this way we exclude saturated pixels, which do not contain useful information, and pixels with little-to-no absorption whose curvature is near zero and uncertain.

\label{fig:Ksim}Curvature calculation for simulated and real spectra. Top two panels: curvature (bottom) from b-spline fits of a simulated, 10 \(h^{-1}\)Mpc-wide spectrum (top) of Ly-\(\alpha\) forest at \(z=2.76\) (blue solid line) and Ly-\(\beta\)\(+\)\(\alpha\) (green solid line). The latter is obtained by contaminating the corresponding Ly-\(\beta\) forest (red dotted line) with a randomly chosen Ly-\(\alpha\) section at lower redshift (black dashed line). Bottom two panels: same as above but for a real Ly-\(\alpha\) and Ly-\(\beta\)\(+\)\(\alpha\) spectrum. The spectra (black lines) are plotted behind the b-spline fits. Shading shows a Ly-\(\beta\)\(+\)\(\alpha\) region contaminated by metal absorption (green dashed line); the corresponding part of the Ly-\(\alpha\) spectrum is also masked (black dashed line).

In Figure \ref{fig:Rg} we present the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relationship obtained by averaging the curvature ratio computed from the synthetic spectra with different thermal histories, i.e. different \(\gamma\) and \(T_{0}\) values. At each redshift, the \(\log\langle R_{\kappa}\rangle\) values computed from each of the simulations (coloured points) and plotted as a function of the \(\gamma\) value characterizing the different thermal histories, lie on the same curve, with little sensitivity to differences in \(T_{0}\).

\label{fig:Rg}Relationship between the slope parameter of the temperature–density relation, \(\gamma\), and the curvature ratio, \(R_{\kappa}\), from our nominal simulations. Each solid line represents this relationship at a particular redshift, obtained from synthetic spectra with different thermal histories. For clarity, we show the values of \(\log\langle R_{\kappa}\rangle\) for the different models, and the relation computed from spectra without noise only for \(z=2.760\) (coloured points and dashed line, respectively). But see Appendix C for the scatter plots corresponding to the other redshifts.

Therefore, for each redshift, it is possible to fit a simple function that connects the mean \(\log\langle R_{\kappa}\rangle\) and \(\gamma\) independently of \(T_{0}\) (see also Appendix C for the complete set of plots). As redshift decreases, the relation shifts towards lower values of \(\log\langle R_{\kappa}\rangle\) and a somewhat steeper relationship with \(\gamma\). Given this tight correspondence, the curvature ratio represents an interesting tool to independently measure the slope of the IGM \(T\)\(\rho\) relation.

The sensitivity of this nominal \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation to the main observational uncertainties in the spectra are tested as follows:

  • Noise: The synthetic spectra from which we obtain the nominal relationship need to include noise at the same level as in the real spectra. If, as an extreme example, no noise was added, Fig. \ref{fig:Rg} shows the effect on the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation: this leads to a \(\sim\)5 per cent decrease in \(\langle R_{\kappa}\rangle\) (and hence a variation of \(\sim 0.03\) dex in \(\log\langle R_{\kappa}\rangle\)).

    This would cause a \(\sim\)8 per cent underestimation of \(\gamma\), comparable to the statistical errors in our sample. Therefore, any errors in how the noise properties are incorporated into the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation should cause relatively small systematic uncertainties in \(\gamma\) measurements of the quality coming from a sample like ours.

  • Optical depth calibration: Changing the effective optical depth used to calibrate the simulations by 10 per cent alters \(\langle R_{\kappa}\rangle\) by only \(\sim\)2 per cent and, consequently, \(\gamma\) by \(\sim\)4 per cent, well within the statistical uncertainties in our sample. This point is particularly promising because the curvature for the Ly-\(\alpha\) forest alone (and the Ly-\(\beta\)\(+\)\(\alpha\) forest alone) is considerably more sensitive to this calibration, as explored in \ref{sec:SimAnalysis}.

However, while the nominal \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation is robust to these observational aspects, uncertainties in the evolution of the IGM thermal state must also be considered (see Section \ref{sec:SystErr}).

The observed curvature ratio


To apply the method to the 27 real quasar spectra we compute the curvature ratio in sections of 10 comoving \(h^{-1}\)Mpc of metal-free Ly-\(\alpha\) and corresponding Ly-\(\beta\)\(+\)\(\alpha\) forest regions. Narrow metal lines (\(b\lesssim\)15 km\(s^{-1}\)) represent a potentially serious source of systematic errors in any measure of forest absorption and should be avoided. With this aim we “clean” the spectra, extending the metal masking procedure described in \ref{sec:metals} to the Ly-\(\beta\)\(+\)\(\alpha\) region: metal absorbers redward of the Lyman-\(\alpha\) emission line are identified and all strong metal transitions at their redshifts are masked out, followed by a by-eye check of the remaining forest.

While the metal correction produces spectra that are reasonably free of contaminants, this procedure reduces the quantity of information available in different sections in a non-uniform way, introducing a possible source of bias. Because the curvature ratio traces differences in the absorption features of two different regions of the same observed spectrum, avoiding systematic effects requires that they cover the same absorption redshift range. Therefore, before measuring \(R_{\kappa}\) from the real data, we mask out regions of the Ly-\(\beta\)\(+\)\(\alpha\) forest corresponding to any range masked from the Ly-\(\alpha\) forest, and vice versa. Figure \ref{fig:Ksim} shows an example of this masking procedure. Finally, possible edge effects are avoided: we do not include the 4 pixels closest to the edge of any masked region in the curvature ratio calculation.

Figure \ref{fig:Ratio} presents the curvature ratio results from our observational sample. The statistical uncertainty in the single \(R_{\kappa}\) measurement, computed from each useful pair of Ly-\(\beta\)\(+\)\(\alpha\) and Ly-\(\alpha\) sections, is negligible compared to the much larger variance among different measurements. Therefore, we collected the measurements in three broad redshift bins which include \(>\)30 individual measurements. In each bin we then calculate the mean \(R_{\kappa}\) and its uncertainty using a bootstrap technique over the individual measurements enclosed. The distribution of the \(\log R_{\kappa}\) values for each bin and the bootstrapped distribution of the \(\log\langle R_{\kappa}\rangle\) value are shown in Figure \ref{fig:bootstrap}.

\label{fig:Ratio}Curvature ratio, \(R_{\kappa}\) from each useful pair of Ly-\(\beta\)\(+\)\(\alpha\) and Ly-\(\alpha\) sections from 27 quasar spectra (black circles). The measurement of \(\log\langle R_{\kappa}\rangle\), averaged within each redshift bin, is shown with 1-\(\sigma\) bootstrap errors both before (red squares) and after (green points) the metal masking procedure. The expected evolution of \(\log\langle R_{\kappa}\rangle\), from our nominal simulations with relatively constant \(\gamma\), is presented for \(\gamma\sim 1.5\) (solid blue line) and \(\gamma\sim 1.0\) (dashed blue line). The redshift bins are \((z_{\rm min},\bar{z},z_{max})=(2.0,2.27,2.5),(2.5,2.86,3.1),(3.1,3.36,3.74)\), and the (metal-masked) \(\log\langle R_{\kappa}\rangle\) measurements are \(-0.110\pm 0.026\), \(-0.019\pm 0.030\) and \(0.017\pm 0.032\), respectively.

\label{fig:bootstrap}Distribution of the \(\log R_{\kappa}\) values for each bin and the bootstrapped distribution of the \(\log\langle R_{\kappa}\rangle\) value. For each of the three redshift bins in which we collected our measurements (figures (a), (b) and (c)) are plotted the distribution of the \(\log R_{\kappa}\) measurements (top panels) and the bootstrapped distribution of the \(\log\langle R_{\kappa}\rangle\) value (bottom panels). Bottom panels: the mean computed directly from the data, without bootstrapping, is indicated with the green solid line and the center value of the bootstrapped distribution of the mean (blue solid line) and its \(1\sigma\) uncertainties (red solid lines) are also shown.

The width and roughly Gaussian shape of the \(\log R_{\kappa}\) distribution within each bin is reproduced by our simulated spectra, providing some confidence that the simulations adequately describe the statistical properties of the observed forest absorption. An example of this agreement is shown in Figure \ref{fig:DisComp} where, for the lowest redshift bin, the observed distribution of \(\log R_{\kappa}\) values is compared with the distribution obtained from one of our simulation models (model F15 in Table \ref{table:simulations}) at \(z=2.355\).

\label{fig:DisComp}Comparison between the shape of the \(\log R_{\kappa}\) distribution observed in the lowest redshift bin (green histogram) and the distribution obtained from the simulation model F15 at \(z=2.355\) (black histogram). Both the distributions have been normalized for the comparison and show similar Gaussian shapes and widths. This agreement give us some confidence that the dispersion of our measurement in Figure \ref{fig:Ratio} is well reproduced by the simulations used in this work.

Figure \ref{fig:Ratio} shows evidence for a mild evolution in \(\langle R_{\kappa}\rangle\) as a function of redshift. A Spearman rank correlation test reveals a positive correlation (\(r\approx 0.26\)) with an associated probability of \(\approx\)0.001. For a constant \(\gamma\), this increase in \(\langle R_{\kappa}\rangle\) is consistent with the expected increase in \(\langle R_{\kappa}\rangle\) at increasing redshifts seen in Fig. \ref{fig:Rg}. In particular, our \(R_{\kappa}\) measurements in Fig. \ref{fig:Ratio} show good agreement with the expected evolution for a model with \(\gamma\sim 1.5\) (blue solid line in Figure  \ref{fig:Ratio}) and this scenario would be in agreement with the recent results of (citation not found: Bolton13) with \(\gamma=1.54\pm 0.11\) at \(z\sim 2.4\). However, the absence of a very large change in the curvature ratio in the redshift range considered is consistent with the assumption in the nominal simulations that \(\gamma\) varies little (\(\la\)0.05) for redshifts \(z=2\)–3.5 (see also Section \ref{sec:SystErr}).

Figure \ref{fig:Ratio} also shows \(\langle R_{\kappa}\rangle\) computed without first masking the metal absorption lines. Even though the effect of metal contamination is important when measuring the curvature of the Ly-\(\alpha\) forest alone (see Section \ref{sec:metals}), it is similar in the corresponding sections of Ly-\(\beta\)\(+\)\(\alpha\) forest, so the curvature ratio is understandably less sensitive to this correction. The results in Fig. \ref{fig:Ratio} show that, even without applying the metal correction, the bias introduced in \(\langle R_{\kappa}\rangle\) is \(\sim\)8 per cent. Therefore, possible errors in the metal masking procedure will introduce negligible uncertainties in \(\langle R_{\kappa}\rangle\) compared to the statistical ones.

Rapid evolution in the IGM thermal state


Our nominal simulations assume only mild evolution in \(T_{0}\) (\(\Delta T_{0}\sim 2000\rm\,K\)) and \(\gamma\) (\(\Delta\gamma\lesssim 0.02\)) between the Ly-\(\beta\) and the foreground Ly-\(\alpha\) redshifts (see Figure 1 in (citation not found: Becker11)). However, even if these are reasonable starting assumptions, if these parameters vary more drastically on short timescales due to, for example, blazar heating ((citation not found: Puchwein12)) or non-equilbrium effects ((citation not found: Puchwein15)) these assumptions may have important consequences for the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation and, therefore, preclude a final conversion of our \(\log\langle R_{\kappa}\rangle\) measurements to \(\gamma\) values with formal error bars.

We can construct a toy model to investigate how sensitive the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation is to rapid evolution in \(T_{0}\) and \(\gamma\) by altering these parameters in the foreground Ly-\(\alpha\) forest via a simple post-processing of the simulated spectra. For a given simulation model, at a given redshift, new synthetic spectra are extracted after imposing a new one-to-one power-law \(T\)\(\rho\) relationship. Note this is an approximation, as it removes the natural dispersion in the temperatures at a given overdensity from shock-heating and/or radiative cooling. However, in this way we can easily modify the \(T_{0}\) and \(\gamma\) parameters, and their evolution, to explore the effect on the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation without running new hydrodynamical simulations.

The largest effects on the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation were found in the following two tests:

  • Rapid evolution in \(T_{0}\): For the redshift range \(z=2\)–3.5, \(T_{0}\) evolves in our nominal simulations such that the temperature difference between the foreground and the Ly-\(\beta\) redshift is small: \(\Delta T_{0}\equiv T_{0}({\rm Ly-}\beta)-T_{0}({\rm foreground})\la 2000\) K. To test the effect of much stronger variations in the temperature at the mean density, we modified the values of \(T_{0}\) in each of our nominal simulations, for the foreground only, using the post-processing approach. We then computed the change in \(\log\langle R_{\kappa}\rangle\) compared to the nominal values, \(\Delta\log\langle R_{\kappa}\rangle\). Fig. \ref{fig:T0var} shows the direct relationship between \(\Delta T_{0}\) and \(\Delta\log\langle R_{\kappa}\rangle\) (black solid line). For example, if \(T_{0}\) changes by a further \(\approx 5000\) K between \(z=3.2\) (Ly-\(\beta\)) and \(z=2.6\) (foreground), we would expect a systematic error in our measurement of \(\log\langle R_{\kappa}\rangle\) of \(\approx\)0.03, similar to the statistical error per redshift bin derived from our 27 quasar spectra. Of course, if we alter \(T_{0}\) by the same amount, with the same post-processing approach, for both the foreground and the Ly-\(\beta\) redshift, the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation does not change.

  • Rapid evolution in \(\gamma\): Similar to the \(\Delta T_{0}\) test above, we emulated rapid changes in \(\gamma\) over short timescales, \(\Delta z\approx 0.6\), by post-processing the foreground Ly-\(\alpha\) only. Figure \ref{fig:gammavar} shows the relationship between the change in \(\gamma\) at the redshift of the foreground Ly-\(\alpha\) forest and \(\Delta\log\langle R_{\kappa}\rangle\) (black solid line); for example, an decrease in the foreground \(\gamma\) by 0.15 implies \(\Delta\log\langle R_{\kappa}\rangle\approx 0.03\), equivalent to the statistical uncertainty in our observations.

\label{fig:T0var}Expected relation between \(\log\langle R_{\kappa}\rangle\) and the temperature change between the redshifts corresponding to the Ly-\(\beta\) and foreground Ly-\(\alpha\) forest (black line; see text for details). The green shading indicates the typical statistical uncertainty in our measured \(\log\langle R_{\kappa}\rangle\) values, per redshift bin. The \(\Delta T_{0}\) in our nominal simulations is typically \(\la\)2000 K (dashed line) and the specific results for the self-consistent ‘T15fast’ and ‘T15slow’ simulations with their statistical errors are shown for \(z=2.7\) and 3.2.

\label{fig:gammavar}Expected relation between \(\log\langle R_{\kappa}\rangle\) and the change in \(\gamma\) at the redshift of the foreground Ly-\(\alpha\) forest (black line; see text for details). The green shading indicates the typical statistical uncertainty in our measured \(\log\langle R_{\kappa}\rangle\) values, per redshift bin. A \(\Delta\gamma=0\) corresponds to the foreground \(\gamma\) value of our nominal simulations (dashed line).

While highlighting the potential importance of assumptions for the foreground Ly-\(\alpha\) forest, the above tests rely on a rather simplistic toy model that does not reproduce self-consistently the evolution of the complex relationships between physical parameters. Ideally, the sensitivity of the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation to different physical assumptions and thermal histories would need to be tested with additional self-consistent simulations in the redshift range of interest. Our simulation suite does offer one such self-consistent test in the case of strong \(T_{0}\) evolution: we used the ‘T15fast’ and ‘T15slow’ simulations of (citation not found: Becker11) (see their Fig. 8) to mimic a possible \(\sim\)5000 K heating event from He ii reionisation at \(z>3\) such that between the Ly-\(\beta\) and foreground redshifts there was a typical temperature decrease of \(\Delta T_{0}\approx 2000\)–4000 K. In both simulations we find that \(\log\langle R_{\kappa}\rangle\) varies by \(\lesssim\)0.01 compared to the nominal simulations (see Fig. \ref{fig:T0var}), which seems less sensitive to substantial evolution in \(T_{0}\) than implied by our simplistic toy model. However, differences in the pressure smoothing scale in these models (which may be acting to improve the agreement) prevent us from reliably estimating the systematic uncertainties involved without further self-consistent tests.

*Moreover, our results in Section \ref{subsec:data} do not support strong \(\gamma\) or \(T_{0}\) evolution with redshift: considering the strong sensitivity, previously explored, to the variation of these parameters, a possible considerable departure from a small evolution, similar to the one assumed in our nominal simulations, would manifest itself in a detectable imprint in the evolution of \(\log\langle R_{\kappa}\rangle\) with redshift that our observations do not show. Using the correlation between the change in \(\gamma\) at the redshift of the foreground Ly-\(\alpha\) forest and \(\Delta\log\langle R_{\kappa}\rangle\) presented in Figure \ref{fig:gammavar}, we can attempt an indicative prediction for the expected \(\log\langle R_{\kappa}\rangle\) evolution in different scenarios of strong evolution in \(\gamma\) with redshift. Figure \ref{fig:gammaMod} shows how considerable departures from just a small evolution in \(\gamma\) would manifest themselves as detectable imprints on the evolution of \(\log\langle R_{\kappa}\rangle\) with redshift, that are inconsistent with our observations. Different toy models, for different scenarios of \(\gamma\) evolution in the redshift range \(z\sim 2.0-4\) are presented in Figure \ref{fig:gammaMod} (a). The corresponding predictions in the \(\log\langle R_{\kappa}\rangle\) are shown in Figure \ref{fig:gammaMod} (b) in comparison with our observational measurements. Our observational results are strongly inconsistent with the models (the “slow evolution” and the “fast evolution”models) which predict a strong decrease in \(\gamma\) (possibly due to the He ii reionization) by more than 0.4, while we cannot exclude a“mild” flattening in the temperature–density relation with a \(\Delta\gamma\sim 0.2\) given the current uncertainties.

\label{fig:gammaMod}Indicative predictions for the \(\log\langle R_{\kappa}\rangle\) evolution with redshift in different scenarios of strong evolution in \(\gamma\). Panel (a): toy models for different scenarios of \(\gamma\) evolution as a function of redshift. The models are indicatively emulating a possible flattening (i.e. a reduction in \(\gamma\)) in the temperature–density relation due to the He ii reionization. Panel (b): predictions for the \(\log\langle R_{\kappa}\rangle\) evolution for the different scenarios of panel (a) (solid and dashed lines), compared with our observational results (green dots). Our observational results are consistent with a small evolution in \(\gamma\) for \(z=2-4\) predicted by our nominal simulation (e.g. solid blue line). Strong evolutionary scenarios in this redshift range with a possible decrease in \(\gamma\) by more than 0.4 (red and green dashed lines) are inconsistent with the observed evolution of \(\log\langle R_{\kappa}\rangle\). Possible systematic uncertainties, arising from assumptions about the evolution of the IGM thermal state, do not allow to exclude possible “mild” flattening in \(\gamma\) (black dashed line).

Despite the above complexities, which require future, more comprehensive simulations to resolve in full, the results of our preliminary analysis suggest that the curvature ratio is a promising alternative tool to measure the density dependence of the IGM thermal state. Our preliminary forward-modeling of the quasar spectra is consistent with \(\gamma\sim 1.5\) and no strong evolution in this quantity at \(2<z<3.5\).

*Jeans smoothing effect

Another potential source of systematic uncertainties may be represented by modifications of the absorbers size during the IGM thermal evolution that can affect the broadening of the absorption lines. This is referred to as the Jeans smoothing effect. To test the sensitivity of the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation to variations in the integrated thermal history we can apply modifications to the density distribution of the absorbers (i.e. overdensity distribution) in the simulations using again a post-processing approach. Firstly, for each simulation model of Table \ref{table:simulations} we impose a one-to-one power-law \(T\)\(\rho\) relationship in which we maintain the original density fields of the nominal simulations. From the \(\log\langle R_{\kappa}\rangle\) values computed from these models we then obtain a “reference” \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation.

Secondly, holding fixed the “instantaneous” values of \(T_{0}\) and \(\gamma\) at each redshift, we substitute the density fields of one nominal simulation to another, evolved with different thermal histories, i.e with different \(T_{0}\) or \(\gamma\). Any difference between the \(\log\langle R_{\kappa}\rangle\) computed from the new extracted spectra and those from the reference ones will then only be due the differences in the overdensity distribution and, consequently, to the Jeans smoothing effect.

Interestingly, Figure \ref{fig:JeansS} shows that the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation is insensitive to variations in the integrated thermal history: if compared with the reference relation (dashed line), in all cases, the values of \(\log\langle R_{\kappa}\rangle\) do not seem to be affected by variations in the overdensity distribution and consequently by the Jeans smoothing effect.

\label{fig:JeansS}Sensitivity of the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation to variations in the integrated thermal history. The \(\log\langle R_{\kappa}\rangle\) values are computed from synthetic spectra obtained after imposing on each simulation a different density field, evolved in models with different \(\gamma\) values (red dots) or different \(T_{0}\) (blue stars). In all cases, the values of \(\log\langle R_{\kappa}\rangle\) do not seem to depart significantly from the reference relation (black dashed line; see text for details).

*The slope of the \(T\)\(\rho\) relation


As shown in Section \ref{sec:SystErr}, the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation \(may\) be sensitive to assumptions about the thermal state of the IGM and its evolution. For this reason we cannot yet present final measurements of \(\gamma\) with formal uncertainties. However, from the analysis presented in this Chapter, the current simulation suite seems to well reproduce the properties of the observed spectra (see Section \ref{subsec:data}) and it is likely that a future refined analysis will provide similar results to the ones that we have currently obtained. In this Section we will therefore present the preliminary measurement of \(\gamma\) obtained using our nominal simulation \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relations (in Figure \ref{fig:Rg}) and the measurements of \(\log\langle R_{\kappa}\rangle\) computed from our 27 quasar spectra (in Figure \ref{fig:Ratio}). Note that the uncertainties presented here are purely statistical and do not include any estimate of the systematic errors stemming from the effects discussed in Section \ref{sec:SystErr}.

Interpolating the observed values of \(\log\langle R_{\kappa}\rangle\) with the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation we obtain our first estimates of \(\gamma\) for redshift \(z=2.0\)–3.8. These results are compared in Figure \ref{fig:gamma} with previous line-fitting measurements from (citation not found: Schaye00), (citation not found: Ricotti00), (citation not found: McDonald01) and (citation not found: Bolton13), and with the recent theoretical models for \(\gamma\)’s evolution presented in (citation not found: McQuinn09) and (citation not found: Puchwein14). Using the 27 lines of sight immediately available, our 1\(\sigma\) uncertainties are comparable with that of the recent measurement by (citation not found: Bolton13) (citation not found: Rudie13) from \(\sim\)6000 individual H i absorbers. Because the line-fitting technique is time-consuming and potentially subjective, our measurement represents a promising improvement. Assuming that the statistical errors dominate the total error budget, combining our 3 binned measurements of \(\gamma\) with a simple weighted mean gives \(\gamma=1.55\pm 0.06\) with little evolution (\(\lesssim\)15 per cent) over \(z=2.0\)–3.8. This result is consistent with the \(\gamma\) value adopted to translate \(T(\bar{\Delta})\) measurements into \(T_{0}\) values in our previous curvature analyses (see Section \ref{sec:T0}).

While these preliminary \(\gamma\) estimates are in good agreement with the (citation not found: Bolton13) result, some tension exists with the low \(\gamma\) values measured by (citation not found: Schaye00), and possibly (citation not found: Ricotti00), in the redshift range \(z=2.5\)–3.0 (but see discussion in (citation not found: Bolton13)). An inverted \(T\)\(\rho\) relation (i.e. \(\gamma\lesssim 1\)) at similar redshifts has also been suggested by analyses of the flux probability distribution ((citation not found: Becker07); (citation not found: Bolton08)) and would correspond to a temperature increase, independent of the density, during He ii reionisation (but see discussion in (citation not found: Becker13)). According to this argument, regions with lower temperature (i.e. lower density) before He ii reionisation would experience a larger temperature increase, manifesting itself as a flattening of the \(T\)\(\rho\) relation (i.e. a decrease in \(\gamma\)). However, in the gas density range traced by the curvature ratio (\(\Delta\gtrsim 2\) for \(z\lesssim 3.8\)) our results are inconsistent at more than 2.5-\(\sigma\) with \(\gamma\lesssim 1\) over the redshift interval considered (again considering only the statistical errors in our \(\gamma\) estimates).

A flattening of the \(T\)\(\rho\) relation (from the asymptotic \(\gamma\approx 1.6\)) is also predicted in the non-equilibrium model of (citation not found: Puchwein14), albeit with only a modest reduction in \(\gamma\) to \(\approx\)1.4. In this simulation, based on the radiative transfer model of (citation not found: HM12), photoionization equilibrium is not assumed during the He ii reionisation phase. A similar dip in \(\gamma\) is also visible in the radiative transfer simulations of (citation not found: McQuinn09) with He ii reionisation at \(z\sim 3\); however, no significant flattening occurs in the case of an early He ii reionisation at \(z\gtrsim 6\). At present, the uncertainties in our measurements do not allow us to distinguish a clear preference for one of the models. We estimate that, according to the statistical uncertainties obtained in this work, a sample of a further \(\sim\)100 quasar spectra covering absorption redshifts 2.7–4.0 would improve the overall statistical precision of the \(\gamma\) measurement to \(\approx\)0.03, allowing a \(\approx\)5-\(\sigma\) discrimination between the \citeauthorMcQuinn09 models. However, such a measurement will only be useful if the possible systematic errors discussed above are resolved and removed in future refinement of the curvature ratio technique.

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We have presented a new method to constrain the \(T\)\(\rho\) relation using the ratio of curvatures, \(\langle R_{\kappa}\rangle\), in the Ly-\(\alpha\) and \(\beta\) forests. The technique allows an independent measurement of \(\gamma\), appears robust against observational uncertainties in the noise level, metal contamination and effective optical depth, and is relatively simple and fast to compute. The \(\langle R_{\kappa}\rangle\) measured in 3 redshift bins covering \(z=2.0\)–3.5 in 27 VLT/UVES quasar spectra, provides \(\approx\)6 per cent statistical uncertainties and matches the redshift evolution derived from our nominal hydrodynamical simulations with \(\gamma\sim 1.5\). In the absence of any other systematics, this statistical error in \(\langle R_{\kappa}\rangle\) translates to a \(\lesssim\)10 per cent uncertainty in \(\gamma\) in \(\Delta z\sim 0.6\) bins. This would be competitive with recent attempts to measure \(\gamma\) using line decomposition (citation not found: Rudie13) (citation not found: Bolton13).

However, we do not present here the \(\gamma\) measurements with formal uncertainties, with any attempted estimate of systematic errors, because the \(\gamma\)\(\log\langle R_{\kappa}\rangle\) relation may also be sensitive to assumptions about the thermal state of the IGM and its evolution. Our nominal simulations predict a decrease of \(\Delta T_{0}\sim 2000\rm\,K\) between the Ly-\(\beta\) and foreground redshifts. If, in reality, there was a further \(\Delta T_{0}\sim 5000\rm\,K\) it would result in a systematic uncertainty in \(\gamma\) which is comparable to the statistical uncertainty. A change of \(\Delta\gamma=0.15\) (cf. \(\Delta\gamma\approx 0.02\) in the nominal simulations) results in a similar uncertainty. Changes of this magnitude may be expected for non-equilibrium photo-heating (citation not found: HM12) (citation not found: Puchwein15) or more exotic models incorporating e.g. blazar heating (citation not found: Puchwein12). Therefore, future efforts should aim to explore the full parameter space for these assumptions. This will likely require a much more comprehensive simulate suite than available for this work so far. Alternatively, these systematics may be largely mitigated if the foreground Ly-\(\alpha\) lines that contaminate the Ly-\(\beta\) absorption can be identified and masked (or divided) out. If this can be achieved and/or the major foreground uncertainties are marginalised over, the curvature ratio promises a novel technique for constraining the evolution of \(\gamma\) over a large redshift range. Distinguishing between models of the IGM’s thermal history could then be improved by utilizing the large samples of high signal-to-noise ratio quasar spectra available in the archives of several high-resolution spectrographs.

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