A New Approach for Constraining Large-Scale Temperature Fluctuations in the Intergalactic Medium

The reionization of helium is thought to occur at $2.5\lesssim z\lesssim4$, marking the last phase transition and final global heating event of the intergalactic medium (IGM). Since it is driven by rare quasars, helium reionization should give rise to strong temperature fluctuations in the IGM between neutral and recently-ionized regions of order $\sigma (\ln T) \sim \Delta T/T = 20-50\%$. We introduce a novel method to search for reionization-induced temperature fluctuations in the IGM by using the effective optical depths of the Lyman-$\alpha$ forest towards a large number of background quasars. Higher IGM temperatures give rise to lower effective optical depths in the Lyman-$\alpha$ forest, implying that temperature fluctuations will broaden the observed optical depth distribution. We measured the distributions of effective Lyman-$\alpha$ forest optical depths across $71$ X-Shooter spectra from the XQ-100 survey in four redshift bins from $z=3.76$ to $z=4.19$ and compared them to a large-volume cosmological hydrodynamical simulation. A good agreement is found between the observations and the simulation, which does not include temperature fluctuations; therefore, we do not detect a signature of helium reionization. We then post-process the simulations to include an increasing amount of temperature fluctuations until the model becomes inconsistent with the observations. We obtain tight constraints on $\sigma (\ln T) < 0.29 \ (<0.40)$ at $2 \sigma\ (3 \sigma)$ at $z=3.76$ when averaging over scales of $100$ comoving Mpc, and weaker constraints for higher redshifts and smaller scales. Our constraints are the tightest to date, and imply that either the IGM temperature contrast caused by helium reionization is less than $\sim30\%$, or that the process has not yet significantly started at $z=3.76$.