Almost all inferences from the Lyman-alpha forest have used the 1D power spectrum.  The reason that the extragalactic astrophysics community places its trust in the power spectrum over other statistics is that various systematics and noise can generally be isolated to the shortest wavelengths.   However, a lot of work has shown that substantial differences from our standard models may go undiscovered in the 1D Lyman-alpha power spectrum -- it is only sensitive to the IGM temperature near a particular density, it is not sensitive to fluctuations in the temperature (or in the radiation background), and  it is not sensitive to processes that affect denser regions.

In Wilson et al. 2022, we investigated the cross-power spectrum between the hydrogen absorption from the Lyman-alpha and Lyman-beta lines.  The hydrogen Lyman-beta absorption owes to higher density gas that lies closer to galaxies because of its five times smaller optical depth relative to Lyman-alpha.    Thus, it is more sensitive to the impact of galaxies on their surrounding gas.  This statistic also enables measuring the density scaling of the temperature, a reflection of heating from the first galaxies.  While the Lyman-beta absorption data is lower quality than that of the Lyman-alpha, the Lyman-alpha--Lyman-beta cross-power immediately removes the sensitivity to any systematic that is uncorrelated between the distinct spectral regions that have the Lyman-alpha and Lyman-beta absorption.  One such systematic is the ubiquitous lines from heavier elements.  This is a significant advantage compared to the Lyman-alpha power spectrum.  

The left panel shows our measurement of this statistic using 100 sightlines through the cosmos taken by the X-Shooter spectrograph on the Very Large Telescope.  The different panels show different cosmic times (indicated by the redshift range, z). The curves show models for the IGM temperature, a property shaped by when the first galaxies and black holes ionized the universe.  The models show larger differences in the cross-power spectrum (green), indicating that this statistic can better differentiate between these models than the standard Lyman-alpha power spectrum (orange).  This statistic also tests the overall IGM paradigm with the consistency test of whether the cross-power measurements agrees with the models that fit the Lyman-alpha power measurements.  Indeed, we found some tension between our past measurement of the cross-power spectrum and the measurement of the Lyman-alpha power spectrum (with our measurement preferring denser regions to be hotter), which may suggest that the models are missing an ingredient.  

One (absolutely terrible!) aspect about  Lyman-alpha forest research is that one typically needs to run large cosmological simulations to do anything. Unlike halos/galaxies for which we have Excursion Set Theory and the Halo Model, permitting some semi-analytic work, the only semi-analytic model that has any success for understanding the forest is a local lognormal mapping of the linear density.  However, once this ansatz is made, one has lost a lot of control.  Vid Irsic and I devised a more controlled  semi-analytic  model, inspired by the Halo Model, for predicting Lyman-alpha forest statistics. This model uses the absorption line decomposition of the forest.  This aspect of the model may hurt its popularity -- cosmologists do not like to think of the forest as discrete lines -- but this decomposition does describe well the properties of the forest.  

The figure on the left (top) shows the z=2 forest, highlighting a number of lines and their columns.  The dotted curve is if you fit all the lines with Gaussians in the optical depth...which essentially produces a perfect fit.  The z<3 forest is, to excellent approximation, a bunch of Gaussian lines.

Our model calculates the statistics of the forest assuming that these lines are Poissonian distributed on small scales and trace the linear density field with some bias on large-scales.  It turns out you can do this entire calculation analytically with these assumptions, resulting in (in my opinion) elegant expressions.  While some might wonder why go through all the trouble since we have simulations, these types of models build intuition into how different pieces fit together and what aspects of the forest measurements are sensitive.  For the most-measured statistic, the 1D Lyman-alpha forest power spectrum, we find that its normalization is set primarily by the number of lines, favoring systems with overdensities of ten, and there is a smaller `two-absorber' component from absorber clustering that derives primarily from overdensities of a few, at least at z=2-3.  The solid colored curves in the panel to left-bottom shows the model predictions (varying the most ill-defined aspect of the model, how absorbers exclude each other at small separations -- although you can also see it makes a modest difference) relative to a simulation (the solid black curve).

When I was starting my postdoc at Berkeley in 2010, it seemed like everyone was trying to derive a formula for the density and velocity gradient biases of the Lyman-alpha forest.  The chance these efforts had to find such a formula was essentially zero as the densities that are seen in the forest are nonlinear.  Our model is the only way to estimate (and understand) these bias coefficients without resorting to simulations.

Lyman-alpha forest analyses often assume a perfect power-law relationship between the temperature and density of gas. (Both the temperature of gas and its density are needed to model the absorption.)  The figure on the left shows what this power-law looks like in a cosmological simulation, where Δ is the density in units of the cosmic mean density.  The figure shows 300 randomly selected particles (black points). Most points lie along a tight relation between temperature and density, with the reason some points deviate because of shocking in the simulation.  Also shown is the distribution  if one turns off shock heating (the red points), which results in the power-law being ridiculously tight!  There is a famous paper by Hui and Gnedin (1997) which solved a linearized equation that includes photoheating and adiabatic cooling, finding an asymptotic power-law index that is consistent with that seen in simulations.   People accepted this work as the final say on the matter and there had not been a paper devoted to understanding this relation since, even though this relation is one of the most striking properties of the IGM, relating the temperature of bulk of the gas in the Universe to its density.  

I was at a conference in Edinburgh in 2012 when the topic came up again.  A discussion broke out about the reason for the relation.  No one could come up with a good explanation (and the one who offered an explanation was not correct). In a paper motivated by this discussion, we showed that more could be added to the Hui and Gnedin discussion of this relation, which had used equations linearized in the matter overdensity (even though the power-law relation holds over two decades in density), ignored all coolants (some of which can be quite important), and did not explain why the relation was so tight (as there are varied paths to reach a given density).  Indeed, the full nonlinear equation can be solved with the most important coolant, Compton cooling off the CMB, and we showed that the solution has almost no dependence on the density evolution of a gas parcel for gas evolving on the timescales expected for gas with densities where this relation applies (Δ < 10) -- explaining why there is so little scatter. We further showed that the effect of other coolants could be added perturbatively.  Finally, we gave an intuitive explanation for the relation:  The heating rate (dominated by photoheating in the standard picture) is such that it is always injecting heat to take gas, essentially independent of density, from one adiabat to the next (and this still holds in presence of Compton cooling).  Adiabatic density evolution also keeps gas on an adiabat.  As a result, a near perfect power-law results.  

The asymptotic T-Δ relation is one part of the story for what shapes the IGM temperature.  While that part of the story is particularly clean, one thing I like about temperature is that (at least in standard picture where only heating process is photoheating), the physics is generally simple. Hydrogen reionization imprints some heating, where the amount of heating depends mostly on how quickly reionization occurs (as discussed here).  Afterwards the gas cools as the Universe expands, with the precise track dependent on well understood atomic cooling physics and the photoheating rate (which depends weakly on the background spectrum).  Later, the second electron of helium is reionized, injecting heat as quasars doubly ionized the helium.  In the standard picture, no other sources of heating are significant.  In a study led by graduate student Phoebe Upton Sanderbeck, we tried to put all of this together and compare with measurements of the IGM temperature.  This is shown in the panels on the left; these minimal models are largely able to explain the temperature at the highest redshifts that have been measured, as well as the amount of heating from HeII reionization at z~3 (which results in the bump in temperature in the plots on the left).  

In 2019, we revisited a key uncertainty in this model, which is especially important for the models at z>5.  The community largely thought that the temperature that the IGM was heated to by reionization was set by the spectrum of the sources.  In a study led by Anson D'Aloisio, we showed that this was not the case, and that it was actually the speed of ionization fronts that establish the temperature.  The faster the ionization front, the hotter the temperature.  Early on in reionization, the fronts move slowly and only heat the IGM to 15,000K, but by the end of reionization as the flux at the bubble edges is set by many sources they move quite fast and can heat it above 30,000K.  This leads to a patchy distribution of temperature that is shown on the left (one panel shows our model based on the speed of ionization fronts and the other a full simulation).  This patchy distribution of temperature also figures into some of my reionization research.